VARIETIES OF DE MORGAN MONOIDS: COVERS OF ATOMS

VARIETIES OF DE MORGAN MONOIDS: COVERS OF ATOMS T. MORASCHINI, J.G. RAFTERY, AND J.J. WANNENBURG Abstract. The variety DMM of De Morgan monoids has just four minimal subvarieties. The join-irreducible covers of these atoms in the subvariety lattice of DMM are investigated. One of the two atoms consisting of idempotent algebras has no such cover; the other has just one. The remaining two atoms lack nontrivial idempotent members. They are generated, respectively, by 4 element De Morgan monoids C 4 and D 4, where C 4 is the only nontrivial 0 generated algebra onto which finitely subdirectly irreducible De Morgan monoids may be mapped by non-injective homomorphisms. The homomorphic pre-images of C 4 within DMM (together with the trivial De Morgan monoids) constitute a proper quasivariety, which is shown to have a largest subvariety U. The covers of the variety V(C 4) within U are revealed here. There are just ten of them (all finitely generated). In exactly six of these ten varieties, all nontrivial members have C 4 as a retract. In the varietal join of those six classes, every subquasivariety is a variety in fact, every finite subdirectly irreducible algebra is projective. Beyond U, all covers of V(C 4) [or of V(D 4)] within DMM are discriminator varieties. Of these, we identify infinitely many that are finitely generated, and some that are not. We also prove that there are just 68 minimal quasivarieties of De Morgan monoids. 1. Introduction De Morgan monoids, introduced by Dunn [7, 22], are involutive distributive residuated lattices satisfying x x 2. The theory of residuated lattices descends from the study of ideal multiplication in rings, and from the calculus of binary relations, but the algebras also model substructural logics; see [11]. In particular, the relevance logic R t of Anderson and Belnap [1] is Key words and phrases. De Morgan monoid, Sugihara monoid, Dunn monoid, residuated lattice, relevance logic. 2010 Mathematics Subject Classification. Primary: 03B47, 06D99, 06F05. Secondary: 03G25, 06D30. This work received funding from the European Union s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 689176 (project Syntax Meets Semantics: Methods, Interactions, and Connections in Substructural logics ). The first author was also supported by RVO 67985807 and by the CAS-ICS postdoctoral fellowship PPLZ 100301751. The second author was supported in part by the National Research Foundation of South Africa (UID 85407). The third author was supported by the DST-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS), South Africa. Opinions expressed and conclusions arrived at are those of the authors and are not necessarily to be attributed to the CoE-MaSS. 1

2 T. MORASCHINI, J.G. RAFTERY, AND J.J. WANNENBURG algebraized by the variety DMM of De Morgan monoids, provided that the monoid identity e is distinguished in the algebraic signature. From the general theory of algebraization [5], it follows that the axiomatic extensions of R t and the subvarieties of DMM form anti-isomorphic lattices, and the latter are susceptible to the methods of universal algebra. Accordingly, in [23], we initiated a study of the lattice of varieties of De Morgan monoids. Among other results, we proved that this lattice has just four atoms. The idempotent De Morgan monoids (a.k.a. Sugihara monoids) are very well understood and encompass two of the minimal varieties, viz. the class BA of Boolean algebras (whose nontrivial members satisfy e < e) and the variety V(S 3 ) generated by the 3 element Sugihara monoid (in which e = e). The remaining two are generated, respectively, by two 4 element algebras C 4 and D 4, where C 4 is totally ordered (with e < e), while e and e are incomparable in D 4. We established in [23] that a subvariety of DMM omits C 4 and D 4 iff it consists of Sugihara monoids. The present paper is primarily an investigation of the covers of these four atoms within DMM. It suffices to consider the join-irreducible covers, as the subvariety lattice of DMM is distributive. We show that BA has no joinirreducible cover within DMM, and that V(S 3 ) has just one; the situation for V(C 4 ) and V(D 4 ) is much more complex (see Theorem 7.2). The covers of V(C 4 ) are distinctive, in view of a result of Slaney [26]: C 4 is the only 0 generated nontrivial algebra onto which finitely subdirectly irreducible De Morgan monoids may be mapped by non-injective homomorphisms. We demonstrate that there is a largest variety U of De Morgan monoids consisting of homomorphic pre-images of C 4 (along with trivial algebras), as well as a largest subvariety M of DMM such that C 4 is a retract of every nontrivial member of M. Thus, V(C 4 ) M U. We furnish U and M with finite equational axiomatizations; each has an undecidable equational theory and uncountably many subvarieties (see Sections 4 and 6). We also provide representation theorems for the members of U and M (Corollaries 5.6 and 5.8), involving a skew reflection construction of Slaney [27]. With the help of these representations, we identify all of the covers of V(C 4 ) within U. There are just ten, of which exactly six fall within M (Theorem 8.10, Corollary 8.14). All ten of these varieties are finitely generated. Within DMM, every cover of V(D 4 ) is semisimple in fact, a discriminator variety. The same applies to the covers of V(C 4 ) that are not contained in U. In both cases, we identify infinitely many such covers that are finitely generated, and some that are not even generated by their finite members (see Sections 9 and 10). In the literature of substructural logics, subvariety lattices are more prominent than subquasivariety lattices, because they mirror the extensions of a logic by new axioms, as opposed to new inference rules. Nevertheless, some natural logical problems call for a consideration of quasivarieties if they are to be approached algebraically, e.g., the identification of the structurally complete axiomatic extensions of R t. Although this particular question is deferred

VARIETIES OF DE MORGAN MONOIDS: COVERS OF ATOMS 3 to a subsequent paper, we throw some fresh light here on the subquasivariety lattice of DMM. Specifically, each of the four minimal varieties of De Morgan monoids is also minimal as a quasivariety, but they are not alone in this. Indeed, we prove that DMM has just 68 minimal subquasivarieties (Corollary 3.6, Remark 5.9). The proof exploits Slaney s description of the free 0 generated De Morgan monoid in [25]. We show, moreover, that in the varietal join J of the six covers of V(C 4 ) within M, every finite subdirectly algebra is projective. It follows that every subquasivariety of J is a variety. (See Theorems 8.12 and 8.13.) 2. Residuated Structures and De Morgan Monoids Some key definitions and results are recalled briefly below. Unproved assertions in this section were either referenced or proved in [23], where additional citations and/or attributions can be found. Familiarity with [23] is not presupposed, however. An involutive (commutative) residuated lattice, or briefly, an IRL, is an algebra A = A;,,,, e comprising a commutative monoid A;, e, a lattice A;, and a function : A A, called an involution, such that A satisfies x = x and (1) x y z z y x. Here, denotes the lattice order and binds more strongly than any other operation; we refer to as fusion. It follows that is an anti-automorphism of A;,, and if we define x y := (x y) and f := e, then A satisfies (2) (3) (4) (5) x y z y x z x = x f, hence x x f, (the law of residuation), x y = y x and x y = (x y), e x = x x x x = x x = x x. An algebra A = A;,,,, e is called a (commutative) residuated lattice or an RL if A;, e is a commutative monoid, A;, is a lattice and is a binary operation called residuation such that A satisfies (2). Every RL satisfies the following well known laws. Here and subsequently, x y abbreviates (x y) (y x). (6) (7) (8) (9) (10) (11) (12) x (x y) y and x (x y) y x y z y x z x (y z) = (x y) (x z) x (y z) = (x y) (x z) (x y) z = (x z) (y z) { x z y z and x y = z x z y and y z x z x y e x y

4 T. MORASCHINI, J.G. RAFTERY, AND J.J. WANNENBURG (13) (14) x = y e x y e x x and e x = x. By (13), an RL A is nontrivial (i.e., A > 1) iff e is not its least element. A class of algebras is said to be nontrivial if it has a nontrivial member. An RL A is said to be bounded if there are extrema, A, by which we mean that a for all a A. In this case, for each a A, (15) a = = and a = = a =. If, moreover, a = for all a A\{ }, then A is said to be rigorously compact [21], in which case a = = b for all a A\{ } and b A\{ }. The extrema of a bounded [I]RL are not distinguished in the algebra s signature, so they are not always retained in subalgebras. Lemma 2.1. Let A be a rigorously compact RL, with extrema,, and let h: A B be a homomorphism that is not a constant function. Then (i) h 1 [{h( )}] = { } and h 1 [{h( )}] = { }. (ii) If h( ) is meet-irreducible in B, then is meet-irreducible in A. Likewise, is join-irreducible if h( ) is. (iii) If B is totally ordered (as a lattice), then is meet-irreducible and join-irreducible in A. Proof. (i) If < a A, with h(a) = h( ), then a =, by rigorous compactness, so h( ) = h( ) h(a) = h( ) h( ) = h( ) = h( ). Similarly, if > b A, with h(b) = h( ), then h( ) = h( ), because b =. As h is isotone, we conclude in both cases that h[a] = 1, contradicting the fact that h is not constant. (ii) follows easily from (i), and (iii) from (ii). In an RL A, we define x 0 := e and x n+1 := x n x for n ω. We say that A is square-increasing if it satisfies x x 2. Every [square-increasing] RL can be embedded into a [square-increasing] IRL, and every finitely generated square-increasing IRL is bounded. The following laws obtain in all squareincreasing IRLs: (16) (17) (18) (19) (20) x y x y x, y e = x y = x y x (x y) x y e x x f x = x 3 = x 2 (in particular, f 3 = f 2 ). An RL A is said to be distributive [resp. modular] if its reduct A;, is a distributive [resp. modular] lattice. Recall that a [quasi]variety is the model class of a set of [quasi-]equations in an algebraic signature. (Quasi-equations have the form (α 1 = β 1 &... & α n = β n ) = α = β,

VARIETIES OF DE MORGAN MONOIDS: COVERS OF ATOMS 5 where n ω.) The class of all RLs and that of all IRLs are finitely axiomatized varieties. They are congruence distributive, congruence permutable and have the congruence extension property (CEP); see [11] for instance. In the lemma below, the acronym [F]SI abbreviates [finitely] subdirectly irreducible. (In any given algebraic signature, the direct product of an empty family is a trivial algebra, hence SI algebras are nontrivial, as are simple algebras.) Every variety is generated by its SI finitely generated members, as Birkhoff s Subdirect Decomposition Theorem [2, Thm. 3.24] says that every algebra is isomorphic to a subdirect product of SI homomorphic images of itself (and since an equation involves only finitely many variables). Lemma 2.2. Let A be a (possibly involutive) RL. (i) A is FSI iff e is join-irreducible in A;,. In this case, therefore, the subalgebras of A are also FSI. (ii) When A is distributive, it is FSI iff e is join-prime (i.e., whenever a, b A with e a b, then e a or e b). (iii) If there is a largest element strictly below e, then A is SI. The converse holds if A is square-increasing. (iv) If e has just one strict lower bound, then A is simple. The converse holds when A is square-increasing. The class of all [F]SI members of a class L of algebras shall be denoted by L [F]SI. The class operator symbols I, H, S, P, P S and P U stand, respectively, for closure under isomorphic and homomorphic images, subalgebras, direct and subdirect products, and ultraproducts, while V and Q denote varietal and quasivarietal generation, i.e., V = HSP and Q = ISPP U = IP S SP U. For each class operator O, we abbreviate O({A 1,..., A n }) as O(A 1,..., A n ). Recall that P U (L) I(L) for any finite set L of finite similar algebras [6, Lem. IV.6.5]. Jónsson s Theorem [16, 18] asserts that, for any subclass L of a congruence distributive variety, V(L) FSI HSP U (L). In particular, if L consists of finitely many finite similar algebras, then V(L) FSI HS(L), provided that V(L) is congruence distributive. Note also that HS(L) = SH(L) for any class L of [I]RLs, owing to the CEP. Corollary 2.3. Let K be any class of simple square-increasing [I]RLs. Then the variety V(K) is semisimple, i.e., its SI members are simple algebras. In fact, its SI members are just the nontrivial algebras in ISP U (K). 1 Proof. By Jónsson s Theorem, the SI members of V(K) belong to HSP U (K), but the criterion for simplicity in Lemma 2.2(iv) is first order-definable and therefore persists in ultraproducts (by Los Theorem [6, Thm. V.2.9]), while the CEP ensures that nontrivial subalgebras of simple algebras are simple. 1 Actually, V(K) is a discriminator variety, so it consists of Boolean products of simple algebras, in the sense of [6, Sec. IV.8 9]. This stronger conclusion will not be needed here, but it follows because the discriminator varieties are just the congruence permutable semisimple varieties with equationally definable principal congruences (EDPC) [4, 9], and because square-increasing [I]RLs have EDPC [11, Thm. 3.55].

6 T. MORASCHINI, J.G. RAFTERY, AND J.J. WANNENBURG An element a of an [I]RL A is said to be idempotent if a 2 = a. We say that A is idempotent if all of its elements are. An IRL is said to be anti-idempotent if it is square-increasing and satisfies x f 2 (or equivalently, (f 2 ) x). This terminology is justified by Theorem 2.4(iii), which implies that a square-increasing IRL A is anti-idempotent iff V(A) has no nontrivial idempotent member. Theorem 2.4. (i) A square-increasing IRL is idempotent iff it satisfies f e, iff it satisfies f 2 = f. Consequently: (ii) A square-increasing non-idempotent IRL has no idempotent subalgebra (and in particular, no trivial subalgebra). (iii) A variety of square-increasing IRLs has no nontrivial idempotent member iff it satisfies x f 2 (i.e., it consists of anti-idempotent algebras). (iv) In a simple anti-idempotent IRL A, if e < a A, then a f = f 2. Proof. (i) (iii) were proved in [23, Thm. 3.3 and Cor. 3.6]. (iv) Let e < a A. By (11), f = e f a f, but by (1), a f f (since a e e), so f < a f. As A is simple and square-increasing, Lemma 2.2(iv) and involution properties show that f has just one strict upper bound in A, which must be f 2, by anti-idempotence. Thus, a f = f 2. Definition 2.5. A De Morgan monoid is a distributive square-increasing IRL. The variety of De Morgan monoids shall be denoted by DMM. A De Morgan monoid satisfies x e iff it is a Boolean algebra (in which the operation is duplicated by fusion). In a partially ordered set P ;, we denote by [a) the set of all upper bounds of an element a (including a itself), and by (a] the set of all lower bounds. If a b P, we use [a, b] to denote the interval {c P : a c b}. If a < b and [a, b] = {a, b}, we say that b covers (or is a cover of) a. Theorem 2.6. Let A be a De Morgan monoid that is FSI. Then (i) A = [e) (f ]; (ii) if A is bounded, then it is rigorously compact. Consequently, (iii) every finitely generated subalgebra of A is rigorously compact. A Sugihara monoid is an idempotent De Morgan monoid; see [1, 8, 12, 13, 24]. The variety SM of all Sugihara monoids coincides with V(S ) for the algebra S = {a : 0 a Z};,,,, 1 on the set of all nonzero integers, where the lattice order is the usual total order, the involution is the usual additive inversion and { the element of {a, b} with the greater absolute value, if a b ; a b = a b if a = b. An IRL is said to be odd if it satisfies f = e. By Theorem 2.4(i), every odd De Morgan monoid is a Sugihara monoid. In the odd Sugihara monoid

VARIETIES OF DE MORGAN MONOIDS: COVERS OF ATOMS 7 S = Z;,,,, 0 on the set of all integers, the operations are defined like those of S, except that 0 takes over from 1 as the neutral element for. The variety of all odd Sugihara monoids is Q(S), whereas SM = Q(S, S ). For each positive integer n, let S 2n denote the subalgebra of S with universe { n,..., 1, 1,..., n} and, for n ω, let S 2n+1 be the subalgebra of S with universe { n,..., 1, 0, 1,..., n}. Note that S 2 is a Boolean algebra. Up to isomorphism, the algebras S n (1 < n ω) are precisely the finitely generated SI Sugihara monoids, whence the algebras S 2n+1 (0 < n ω) are just the finitely generated SI odd Sugihara monoids. The algebra S 3 is a homomorphic image of S n for all integers n 3. Thus, every nontrivial variety of Sugihara monoids includes S 2 or S 3. Theorem 2.7. (i) ([24, 12]) Every quasivariety of odd Sugihara monoids is a variety. (ii) The lattice of varieties of odd Sugihara monoids is the chain V(S 1 ) V(S 3 ) V(S 5 )... V(S 2n+1 )... V(S). An algebra is said to be n generated (where n is a cardinal) if it has a generating subset with at most n elements. Thus, an IRL is 0 generated iff it has no proper subalgebra. We depict below the two-element Boolean algebra 2 (= S 2 ), the three-element Sugihara monoid S 3, and two four-element De Morgan monoids, C 4 and D 4. In each case, the labeled Hasse diagram determines the structure. e 2: S 3 : e = f f C 4 : f 2 f e (f 2 ) D 4 : 2 f e f (f 2 ) Theorem 2.8. A De Morgan monoid is simple and 0 generated iff it is isomorphic to 2 or to C 4 or to D 4. Lemma 2.9. Let A be a nontrivial square-increasing IRL, and K a variety of square-increasing IRLs. (i) If A is anti-idempotent, with e f, then e < f. (ii) If e < f in A, then C 4 can be embedded into A. (iii) If A is simple and C 4 or D 4 can be embedded into A, then A is anti-idempotent. (iv) If C 4 can be embedded into every SI member of K, then K consists of anti-idempotent algebras and satisfies e f. (v) If D 4 can be embedded into every SI member of K, then K consists of anti-idempotent algebras. Proof. (i) and (ii) were proved in [23, Sec. 5]. (iii) follows from Lemma 2.2(iv), because (f 2 ) < e in C 4 and in D 4. (iv) Suppose C 4 embeds into every SI member of K. Then K satisfies e f, as C 4 does. Now let B K be nontrivial. Then B IP S {B i : i I}

8 T. MORASCHINI, J.G. RAFTERY, AND J.J. WANNENBURG for suitable SI algebras B i K, by the Subdirect Decomposition Theorem. As C 4 embeds into each B i, it embeds diagonally into i I B i, and therefore into B, because it is 0 generated. Thus, no nontrivial B K is idempotent, and so K satisfies x f 2, by Theorem 2.4(iii). The proof of (v) is similar. The next lemma generalizes [26, Thms. 2, 3] (where it was confined to FSI De Morgan monoids). Lemma 2.10. Let A be a rigorously compact IRL. (i) There is at most one homomorphism from A into C 4. (ii) If there is a homomorphism from A to C 4, then (f 2 ) a f 2 for all a A. Proof. Let, be the extrema of A. Suppose h 1, h 2 : A C 4 are homomorphisms, and note that they are surjective, because C 4 is 0 generated. For each i {1, 2}, as h i is isotone and preserves,, e, we have h i (f 2 ) = f 2 = h i ( ) and h i ( (f 2 )) = (f 2 ) = h i ( ), so by Lemma 2.1(i), f 2 = and (f 2 ) = (proving (ii)) and (21) h 1 i [{f 2 }] = {f 2 } and h 1 i [{ (f 2 )}] = { (f 2 )}. Therefore, if h 1 h 2, then h 1 (a) = e and h 2 (a) = f for some a A. In that case, h 2 (a 2 ) = f 2, so a 2 = f 2 (by (21)), whence h 1 (a 2 ) = f 2, contradicting the fact that h 1 (a 2 ) = (h 1 (a)) 2 = e 2 = e. Thus, h 1 = h 2, proving (i). Theorem 2.11. (Slaney [26, Thm. 1]) Let h: A B be a homomorphism, where A is an FSI De Morgan monoid, and B is nontrivial and 0 generated. Then h is an isomorphism or B = C 4. 3. Minimality The following general result will be needed in our study of the subvariety lattice of DMM. Theorem 3.1. ([17, Cor. 4.1.13]) If a nontrivial algebra of finite type is finitely generated, then it has a simple homomorphic image. A quasivariety is said to be minimal if it is nontrivial and has no nontrivial proper subquasivariety. If we say that a variety is minimal (without further qualification), we mean that it is nontrivial and has no nontrivial proper subvariety. When we mean instead that it is minimal as a quasivariety, we shall say so explicitly, thereby avoiding ambiguity. Recall that V(2) is the class of all Boolean algebras. Theorem 3.2. ([23, Thm. 6.1]) The distinct classes V(2), V(S 3 ), V(C 4 ) and V(D 4 ) are precisely the minimal varieties of De Morgan monoids.

VARIETIES OF DE MORGAN MONOIDS: COVERS OF ATOMS 9 A variety K is said to be finitely generated if K = V(A) for some finite algebra A (or equivalently, K = V(L) for some finite set L of finite algebras). Every finitely generated variety is locally finite, i.e., its finitely generated members are finite algebras [6, Thm. II.10.16]. Bergman and McKenzie [3] showed that every locally finite congruence modular minimal variety is also minimal as a quasivariety, so by Theorem 3.2, V(2), V(S 3 ), V(C 4 ) and V(D 4 ) are minimal as quasivarieties. We proceed to show that the total number of minimal subquasivarieties of DMM is still finite, but much greater than four. Lemma 3.3. Let A and B be nontrivial algebras, where A is 0 generated. (i) If B Q(A), then A can be embedded into B, whence Q(A) = Q(B). (ii) Q(A) is a minimal quasivariety. (iii) If B Q(A) and B is 0 generated, then A = B. (iv) If A has finite type and Q(A) is a variety, then A is simple. Proof. (i) Let B Q(A) = ISPP U (A). Then B embeds into a direct product D of ultrapowers of A, where the index set of the direct product is not empty (because B is nontrivial). Clearly, if a variable-free equation ε is true in A, then it is true in B. Conversely, if ε is true in B, then it is true in D, as variable-free equations persist in extensions (i.e., super-algebras). In that case, since ε persists in homomorphic images, it is true in an ultrapower U of A, whence it is true in A, because all first order sentences persist in ultraroots. There is therefore a well defined injection k : A B, given by α A (c A 1, c A 2,... ) α B (c B 1, c B 2,... ), where c 1, c 2,... are the nullary operation symbols of the signature and α is any term. Clearly, k is a homomorphism from A into B, so A IS(B). (ii) follows immediately from (i). (iii) In the proof of (i), the image of the embedding k is a subalgebra of B. So, if B is 0 generated, then k is surjective, i.e., k : A = B. (iv) Suppose A has finite type and is not simple. As A is 0 generated and nontrivial, it has a simple homomorphic image C, by Theorem 3.1, and C is still 0 generated. If C Q(A), then A = C, by (iii), contradicting the non-simplicity of A. So, C / Q(A), whence Q(A) is not a variety. Theorem 3.4. A quasivariety of De Morgan monoids is minimal iff it is V(S 3 ) or Q(A) for some nontrivial 0 generated De Morgan monoid A. Proof. Sufficiency follows from Lemma 3.3(ii) and previous remarks about V(S 3 ). Conversely, let K be a minimal subquasivariety of DMM. Being minimal, K is Q(A) for some nontrivial De Morgan monoid A. Let B be the smallest subalgebra of A. If B is trivial, then A satisfies e = f, so K is a variety, by Theorems 2.4(i) and 2.7(i). In this case, as K is a minimal variety of odd Sugihara monoids, it is V(S 3 ), by Theorem 2.7(ii). On the other hand, if B is nontrivial, then K = Q(B) (again by the minimality of K), and this completes the proof, because B is 0 generated.

10 T. MORASCHINI, J.G. RAFTERY, AND J.J. WANNENBURG A deductive filter of a (possibly involutive) RL A is a lattice filter G of A;, that is also a submonoid of A;, e. Thus, [e) is the smallest deductive filter of A. The lattice of deductive filters of A and the congruence lattice Con A of A are isomorphic. The isomorphism and its inverse are given by G ΩG := { a, b A 2 : a b G}; θ {a A: a e, e θ}. For a deductive filter G of A and a, b A, we often abbreviate A/ΩG as A/G, and a/ωg as a/g, noting that a b G iff a/g b/g in A/G. In the square-increasing case, the deductive filters of A are just the lattice filters of A;, that contain e, by (16), so [b) is a deductive filter whenever e b A, and if A is finite, then all of its deductive filters have this form. For any quasivariety K and any cardinal m, the free m generated algebra in K shall be denoted by F K (m) if it exists (i.e., if m > 0 or the signature of K includes a constant symbol). Theorem 3.5. The minimal subquasivarieties of DMM form a finite set, whose cardinality is the number of lower bounds of e in F DMM (0). Proof. Let F = F DMM (0). Slaney [25] proved that F has just 3088 elements; its bottom element is e F f F (see [23, Thm. 3.2]). By the Homomorphism Theorem, every 0 generated De Morgan monoid is isomorphic to a factor algebra of F, so DMM has only finitely many minimal subquasivarieties, by Theorem 3.4. Now consider a factor algebra F /G, where G is a deductive filter of F. As F is finite, G = [α F ) for some nullary term α in the language of IRLs, where α F e F. If F /G is nontrivial, i.e., α F e F f F, then F /G is not odd (by (13)), whence Q(F /G) V(S 3 ). The function α F Q(F /[α F )) is therefore a well defined surjection from the lower bounds of e F in F to the set consisting of the trivial subvariety (corresponding to the bottom element of F ) and the minimal subquasivarieties of DMM, other than V(S 3 ). It remains only to show that this map is injective. To that end, suppose F /[α F ) and F /[β F ) generate the same quasivariety, where α F, β F e F. Then there is an isomorphism g : F /[α F ) = F /[β F ), by Lemma 3.3(iii). As β F e F, we have β F e F = β F [β F ), by (12) and (14), so β F /[βf ) = e F /[βf ). Now g(β F /[α F )) = g(β F /[αf ) ) = β F /[βf ) = e F /[βf ) = g(e F /[α F )), but g is injective, so β F /[α F ) = e F /[α F ), i.e., β F = β F e F [α F ). This means that α F β F and, by symmetry, α F = β F, completing the proof. Corollary 3.6. There are exactly 68 minimal quasivarieties of De Morgan monoids. Proof. By Theorem 3.5, we need to show that e has just 68 lower bounds in F DMM (0). The argument will be given in Remark 5.9, after the notion of a skew reflection has been defined.

VARIETIES OF DE MORGAN MONOIDS: COVERS OF ATOMS 11 4. Crystalline Varieties We begin this section with some general observations about retracts, that will be needed later. Recall that an algebra A is said to be a retract of an algebra B if there are homomorphisms g : A B and h: B A such that h g is the identity function id A on A. This forces g to be injective and h surjective; we refer to h as a retraction (of B onto A). The composite of two retractions, when defined, is clearly still a retraction. Remark 4.1. Given similar algebras A and B, the first canonical projection π 1 : A B A is a retraction iff there exists a homomorphism f : A B. (Sufficiency: as id A and f are homomorphisms, so is the function g from A to A B defined by a a, f(a), and clearly π 1 g = id A.) Consequently, if an algebra C is a retract of every member of a class K, then D is a retract of D E for all D, E K, because there is always a composite homomorphism from D to E (whose image is isomorphic to C). Remark 4.2. A 0 generated algebra A is a retract of an algebra B if there exist homomorphisms g : A B and h: B A. For in this case, every element of A has the form α A (c 1,..., c n ) for some term α and some distinguished elements c i A, whence h g = id A, because homomorphisms preserve distinguished elements (and respect terms). Lemma 4.3. Let K be a variety of finite type, and let A K be finite, simple and 0 generated. Then the following conditions are equivalent. (i) A is a retract of every nontrivial member of K. (ii) Every simple algebra in K is isomorphic to A and embeds into every nontrivial member of K. Proof. (i) (ii): For each simple C K, there is a homomorphism h from C onto A, by (i), and h must be an isomorphism (as A is nontrivial and C is simple). Thus, the embedding claim also follows from (i). (ii) (i): By (ii) and Theorem 3.1, A is a homomorphic image of every finitely generated nontrivial member of K. Consider an arbitrary nontrivial algebra B K. By (ii), A IS(B). Like any nontrivial algebra, B embeds into an ultraproduct U of finitely generated nontrivial subalgebras B i of B (cf. [6, Thm. V.2.14]). As A H(B i ) for all i, and as P U H(L) HP U (L) for any class L of similar algebras, there is a homomorphism h from U onto an ultrapower of A. But A, being finite, is isomorphic to all of its ultrapowers, so h restricts to a homomorphism from B into A. Therefore, A is a retract of B, by Remark 4.2. Generalizing the usage of [26], we say that an IRL A is crystalline if there is a homomorphism h: A C 4 (in which case h is surjective). 2 Theorem 2.11 motivates the following definitions. 2 For the sake of Theorem 4.5, we have dropped the requirement in [26] that crystalline algebras be FSI.

12 T. MORASCHINI, J.G. RAFTERY, AND J.J. WANNENBURG Definition 4.4. (i) W := {A DMM : A = 1 or A is crystalline}; (ii) N := {A DMM : A = 1 or C 4 is a retract of A} W. By Lemma 2.10(ii), the rigorously compact algebras in W are anti-idempotent. Also, A is a retract of A B for all nontrivial A, B N, by Remark 4.1. Theorem 4.5. W and N are quasivarieties. Proof. As W and N are isomorphically closed, we must show that they are closed under S, P and P U, bearing Remark 4.2 in mind. If B S(A) and h: A C 4 is a homomorphism, then so is h B : B C 4, while any embedding C 4 A maps into B, as C 4 is 0 generated. Thus, W and N are closed under S. Let {A i : i I} be a subfamily of W, where, without loss of generality, I. For any j I, the projection i I A i A j can be composed with a homomorphism A j C 4, so i I A i W. If, moreover, A i N for all i, then C 4 embeds diagonally into i I A i, whence i I A i N. Every ultraproduct of {A i : i I} can be mapped into C 4, as in the proof of Lemma 4.3 ((ii) (i)). Also, as C 4 is finite and of finite type, the property of having a subalgebra isomorphic to C 4 is first order-definable and therefore persists in ultraproducts. Thus, W and N are closed under P and P U. Nevertheless, W and N are not varieties, i.e., they are not closed under H. To see this, consider any simple De Morgan monoid A of which C 4 is a proper subalgebra, and let B = C 4 A. Then B N, by Remark 4.1. Now A H(B) but A / W, because A is simple and not isomorphic to C 4. Concrete examples of finite simple 1 generated De Morgan monoids having C 4 as sole proper subalgebra are given in Section 9. An [I]RL is said to be semilinear if it is a subdirect product of totally ordered algebras. The semilinear De Morgan monoids are axiomatized, relative to DMM, by e (x y) (y x) [15]. The examples in Section 9 show that even the semilinear anti-idempotent algebras in W or N do not form a variety. Note that N contains (semilinear) algebras that are not anti-idempotent. For instance, C 4 C # 4 N does not satisfy x f 2, where C # 4 denotes the rigorously compact extension of C 4 by new extrema,. As W and N are not varieties, it is not obvious that either of them possesses a largest subvariety, but we shall show that both do. Purely equational axioms will be needed in the proof, and the opaque postulate (24), which abbreviates an equation, is introduced below for that reason. The following convention helps to eliminate some burdensome notation. Convention 4.6. In an anti-idempotent IRL, we define 1 := f 2 and 0 := 1 = (f 2 ). (These abbreviations will be used when they enhance readability, rather than always. The typeface distinguishes them from standard uses of 0, 1.)

VARIETIES OF DE MORGAN MONOIDS: COVERS OF ATOMS 13 Definition 4.7. We denote by U the variety of De Morgan monoids satisfying (22) (23) (24) x 2 ( x) 2 = 1 1 (x y) (1 x) (1 y) 1 x y q(x) q(y) q(x y) q(x y) q(x y) (1 (x y)), where q(x) := 1 ( x) 2. (Note that U consists of anti-idempotent algebras, by (22), so our use of the symbol 1 in this definition is justified.) Lemma 4.8. Every rigorously compact member of W belongs to U. Proof. Let A W be rigorously compact. We may assume that A is nontrivial, so there is a (surjective) homomorphism from A to C 4. Because C 4 satisfies (22), [1 (x y)] [(1 x) (1 y)] = 1 and [1 x y q(x) q(y)] [q(x y) q(x y) q(x y) (1 (x y))] = 1, it follows from Lemma 2.1(i) that A satisfies the same laws. Then A satisfies (23) and (24), by (12). Thus, A U. Theorem 4.9. U is the largest subvariety of W, i.e., U is the largest variety of crystalline (or trivial) De Morgan monoids. Proof. To see that U W, let A U be SI. It suffices to show that A W, because W, like any quasivariety, is closed under IP S. Now A is nontrivial and bounded by 0, 1, so 0 < e 1 and A is rigorously compact, by Theorem 2.6(ii). It follows from (23), Lemma 2.2(ii) and (12) that 1 is joinirreducible (whence 0 is meet-irreducible) in A. Let B = {a A : a 0 and ( a) 2 = 1 } and B = {a A : a B}. Then e B (by definition of 1 ) and 1 / B (as 0 2 = 0 1 ), so e < 1. We claim that B is closed under the operations,,, of A. Indeed, let b, c B, so 0 < b, c A and ( b) 2 = 1 = ( c) 2, i.e., q(b) = 1 = q(c). Then b c 0 and ( (b c)) 2 = 1 (because ( (b c)) 2 ( b) 2 ), so b c B. Clearly, b c 0. Also, b c 0, by (16), and 1 b c q(b) q(c) = 1, by rigorous compactness. Then, by (24), each of q(b c), q(b c), q(b c) and 1 (b c) is 1. Thus, 1 = ( (b c)) 2 = ( (b c)) 2 = ( (b c)) 2, again by rigorous compactness, and b c 0. This shows that b c, b c, b c B, as claimed. Let a A\{0, 1 }. Since 1 is join-irreducible, (22) shows that a B or a B, i.e., a B B. Suppose a B B, i.e., a, a B. Then a a = (( a) 2 ) = 1 (as a B) = 0, so ( a a) 2 = 0 1, so a a / B, contradicting the fact that B is closed under. Therefore, A is the disjoint union of B, B, {0 } and {1 }. Suppose b, c B, with c b. Then b 1, so b 0 and b 2 ( c) 2 = 1, so b 2 = 1, hence b B, i.e., b B B =, a contradiction. Thus, no element of B has a lower bound in B. This, together with the meet- [resp.

14 T. MORASCHINI, J.G. RAFTERY, AND J.J. WANNENBURG join-] irreducibility of 0 [resp. 1 ], shows that b d B and b d B for all b B and d B. Let h: A C 4 be the function such that h(0 ) = 0 and h(1 ) = 1 and h(b) = e and h( b) = f for all b B. It follows readily from the above conclusions that h is a homomorphism from A to C 4, so A W, as required. Finally, let K be a subvariety of W. The finitely generated SI algebras in K are rigorously compact, by Theorem 2.6(iii), so they belong to U, by Lemma 4.8. Thus, K U. Remark 4.10. In C 4, we have f a = 0 iff a {0, e}, while a e = 0 iff a {f, 1 }. Therefore, C 4 satisfies (f x) (x e) 0, and hence also (25) ((f x) (x e)) 0 = 0. So, because every SI homomorphic image of a member of U is rigorously compact and crystalline, it follows from Lemma 2.1(i) that U satisfies (25). Note that N and W do not satisfy (25), as (25) fails in the algebra C 4 C # 4 mentioned before Convention 4.6. Definition 4.11. We denote by M the variety of anti-idempotent De Morgan monoids satisfying e f and (25). Lemma 4.12. C 4 is a retract of every nontrivial member of M. Proof. Because M satisfies e f, it also satisfies (26) x f x, and therefore (27) e x = f x f x. As M satisfies (25) and 0 0 = 1, its nontrivial members satisfy (f x) (x e) 0, i.e., (f x) (f x) 0, or equivalently (by De Morgan s laws), (28) (f x) (f x) 1. By Lemma 2.9(i),(ii), every nontrivial member of M satisfies e < f and has a subalgebra isomorphic to C 4. So, by Lemma 4.3, it suffices to show that every simple member of M is isomorphic to C 4. Suppose A M is simple. We may assume that C 4 S(A). We claim that the intervals [0, e], [e, f] and [f, 1 ] of A are doubletons, i.e., (29) [0, e] = {0, e} and [e, f] = {e, f} and [f, 1 ] = {f, 1 }. The first and third assertions in (29) follow from Lemma 2.2(iv) and involution properties. To prove the middle equation, suppose a A with e < a < f. As f = e, it follows that e < a < f and, by (27), f = f a f a. As e a e, we have f a f (by (1)), so f < f a. Then f a = 1, as [f, 1 ] = {f, 1 }. By symmetry, f a = 1, so (f a) (f a) = 1, contradicting (28). Therefore, [e, f] = {e, f}, as claimed.

VARIETIES OF DE MORGAN MONOIDS: COVERS OF ATOMS 15 To complete the proof, it suffices to show that every element of A is comparable with e, as that will imply, by involution properties, that every element is comparable with f, forcing A = {0, e, f, 1 } = C 4. Suppose, on the contrary, that a A is incomparable with e, i.e., a is incomparable with f. As a e and a f, we have e < e a and f < f a, as well as e a (by Theorem 2.6(i)), i.e., a f. So, by (27), f a f a, hence f < f a, and so f a = 1, because [f, 1 ] = {f, 1 }. Again, as e a e, we have f a f, so e f a, by Theorem 2.6(i). This, with e < f, gives e f (f a). Also, a f a, by (26), so a f (f a). Therefore, e a f (f a). If we can argue that f (f a) < f, then e < e a < f, contradicting the fact that [e, f] = {e, f}. So, to finish the proof, it suffices to show that f is incomparable with f a, and we have already shown that f a f. If f < f a, then f a = 1, as [f, 1 ] = {f, 1 }, but since f a = 1, this yields (f a) (f a) = 1, contradicting (28). Therefore, f and f a are indeed incomparable, as required. Theorem 4.13. M is the largest subvariety of N. Proof. By Lemma 4.12, M is a subvariety of N. Let K be any subvariety of N. Clearly, K satisfies e f and, by Lemma 2.9(iv), its members are antiidempotent. Now K is a subvariety of U, by Theorem 4.9, because N W. By Remark 4.10, (25) is satisfied by U, so it holds in K. Thus, K M. Corollary 4.14. M is the class of all algebras in U satisfying e f. In particular, M satisfies (22), (23) and (24). Corollary 4.15. Every rigorously compact algebra in N belongs to M. Proof. This follows from Lemma 4.8 and Corollary 4.14. At this point in our account, N and M are organizational tools, suggested by Theorem 2.11. They will assume an additional significance when we discuss structural completeness in a subsequent paper. (The structurally complete varieties of De Morgan monoids fall into two classes a denumerable family that is fully understood, and a more opaque family of subvarieties of M.) 5. Skew Reflections and U We are going to provide a representation theorem for algebras in U, using ideas of Slaney [27]. 3 Definition 5.1. Let B = B; B, B, B, B, e be a square-increasing RL, with lattice order B. Let B = {b : b B} be a disjoint copy of the set B, let 0, 1 be distinct non-elements of B B, and let S = B B {0, 1 }. Let be a binary relation on S such that 3 The nomenclature of [27] is untypical. There, De Morgan monoids were not required to be distributive, and likewise the Dunn monoids of Definition 5.5.

16 T. MORASCHINI, J.G. RAFTERY, AND J.J. WANNENBURG (i) is a lattice order whose restriction to B 2 is B (the meet and join operations of S; being denoted by and, respectively), and for all b, c B, (ii) b c iff c b, (iii) b c iff e (b B c), (iv) b c, (v) 0 b 1 and 0 b 1. The skew reflection S (B) of B is the algebra S;,,,, e such that (vi) is a commutative binary operation on S, extending B, (vii) a 0 = 0 for all a S, and if 0 a S, then a 1 = 1, (viii) b c = (b B c) and b c = 1 for all b, c B, (ix) 0 = 1 and 1 = 0 and b = b and (b ) = b for all b B. A skew reflection of B is any algebra of the form S (B), where is a binary relation on S satisfying (i) (v). (Some examples are pictured before Lemma 8.6.) Definition 5.1 is essentially due to Slaney [27]. (In [27], (iii) is formulated in an ostensibly more general manner, as for all a, b, c B, we have a B b c iff a (b B c). This follows from (iii), however. Indeed, for a, b, c B, a B b c iff e ((a B b) B c) = (a B (b B c)) iff a (b B c).) By an RL subreduct of an IRL A = A;,,,, e, we mean a subalgebra of the RL reduct A;,,,, e of A. Theorem 5.2. ([27, Fact 1]) A skew reflection S (B) of a square-increasing RL B is a square-increasing IRL, and B is an RL subreduct of S (B). Remark 5.3. In a skew reflection S (B) of a square-increasing RL B, we have f = e, hence f 2 = 1, so S (B) is anti-idempotent and our use of 0, 1 in Definition 5.1 is consistent with Convention 4.6. By definition, S (B) is rigorously compact. Because it has B as an RL subreduct, S (B) satisfies (f x) (x e) 0, and hence also (25). It satisfies (22) and (24) as well. (In verifying (24), we may assume that its left-hand side is not 0, whence x, y, q(x), q(y) 0. This forces x, y B, whence each conjunct of the righthand side is 1.) The fact that elements of B lack lower bounds in B has two easy but important consequences. First, S (B) is simple iff B is trivial (i.e., e is the least element of B), in view of Lemma 2.2(iv). Secondly, by Lemma 2.2(iii), S (B) is SI iff B is SI or trivial. Specifically, when B is not trivial, an element of S (B) is the greatest strict lower bound of e in S (B) iff it is the greatest strict lower bound of e in B. Elements of B might lack upper bounds in B, e.g., D 4 arises in this way from a trivial RL. Such cases are eliminated in the next theorem, however.

VARIETIES OF DE MORGAN MONOIDS: COVERS OF ATOMS 17 Theorem 5.4. The following two conditions on a square-increasing IRL A are equivalent. (i) There is a homomorphism h: A C 4 and A is rigorously compact. (ii) A is a skew reflection of a square-increasing RL B, and 0 is meetirreducible in A. In this case, in the notation of Definition 5.1, (iii) h is unique and surjective, and 1 is join-irreducible in A; (iv) b c B and b c B for all b, c B, so each element of B has an upper bound in B, and elements of B have lower bounds in B; (v) if B is distributive and A is modular, then A is distributive and therefore a De Morgan monoid, belonging to U. Proof. Note first that, in (iii), the uniqueness of h follows from Lemma 2.10(i) (and its surjectivity from the fact that C 4 is 0 generated). (i) (ii): Being crystalline, A is nontrivial. The set B := h 1 [{e}] is the universe of an RL subreduct B of A, which inherits the square-increasing law, and b b := b defines an antitone bijection from B onto B := h 1 [{f}]. Clearly, B B = and no element of B is a lower bound of an element of B, because h is isotone and e < f in C 4. As h fixes 0 and 1, Lemma 2.1 shows that A is anti-idempotent, with h 1 [{0 }] = {0 } and h 1 [{1 }] = {1 }, and that 0 [resp. 1 ] is meet- [resp. join-] irreducible in A, finishing the proof of (iii). In particular, A = B B {0 } {1 } (disjointly). We verify that A satisfies conditions (iii) and (viii) of Definition 5.1. Let b, c B. Because B is closed under the operation of A, we have b c iff b e c iff b c f (by (1), deployed in A), iff e (b c). Clearly, b c = (b c) and h(b c ) = h(b) h(c) = f 2 = 1, so b c = 1. This completes the proof that A = S (B), where is the lattice order of A. (ii) (i): Rigorous compactness was noted in Remark 5.3. Definition 5.1 shows that, and e are preserved by the function h: A C 4 such that h(0 ) = 0, h(1 ) = 1, h(b) = e and h(b ) = f for all b B. As 0 is meetirreducible (whence 1 is join-irreducible) in A, the map h preserves, too. Indeed, if b, c B, then b b c 0 and b has no lower bound in B, so b c B and, by involution properties, b c B. This proves (i) and (iv). By (iv), when S (B) is modular, it will be distributive iff the five-element lattice with three atoms doesn t embed into the sublattice B B of S (B); see [6, Thms. I.3.5, I.3.6]. That is true if B is distributive, as B and B are then distributive sublattices of B B. This, with Lemma 4.8, proves (v). Definition 5.5. A Dunn monoid is a square-increasing distributive RL. Dunn monoids originate in [7] and acquired their name in [19]. Corollary 5.6. A De Morgan monoid belongs to U iff it is isomorphic to a subdirect product of skew reflections of Dunn monoids, where 0 is meetirreducible in each subdirect factor.

18 T. MORASCHINI, J.G. RAFTERY, AND J.J. WANNENBURG Proof. The forward implication follows from Theorem 5.4 and the Subdirect Decomposition Theorem, because the SI homomorphic images of members of U are bounded by 0, 1, are rigorously compact (Theorem 2.6(ii)) and are still crystalline (U being a variety), and because RL subreducts of De Morgan monoids inherit distributivity. Conversely, by Remark 5.3, skew reflections of Dunn monoids satisfy the defining postulates of U, except possibly for (23) and distributivity (which are effectively given here), and U, like any quasivariety, is closed under IP S. Lemma 5.7. Let A = S (B) be a skew reflection of a square-increasing RL B, where A satisfies e f. Then, in the notation of Definition 5.1, (i) b (b e) for all b B, and (ii) 0 is meet-irreducible and 1 is join irreducible in A. Proof. (i) Let b B. By (6), b (b e) e, so e f (b (b e)). Then b (b e), by Definition 5.1(iii). (ii) Let b, c B. By (i), c (c e), i.e., c e c. Because B is an RL subreduct of A and 0 / B, we have b c b (c e) B, so b c 0. As B and B are both sublattices of A, this shows that 0 is meet-irreducible (whence 1 is join-irreducible) in A. Corollary 5.8. A De Morgan monoid belongs to M iff it satisfies e f and is isomorphic to a subdirect product of skew reflections of Dunn monoids. Proof. This follows from Lemma 5.7(ii) and Corollaries 4.14 and 5.6. Remark 5.9. We can now complete the proof of Corollary 3.6. In [25], Slaney showed that the free 0 generated De Morgan monoid F is 2 D 4 A, where A is a skew reflection of the direct product of four Dunn monoids, called the α segments of CA6, CA10a, CA10b and CA14. In each of 2, D 4 and the four α segments, e has just one strict lower bound. So, from the structure of skew reflections, it follows that the number of lower bounds of e F in F (including e F itself) is 2 2 ((2 2 2 2) + 1) = 68. 6. Reflections and M Definition 6.1. Let B be a square-increasing RL, with lattice order B, and let S = B B {0, 1 }, where B = {b : b B} is a disjoint copy of B and 0, 1 are distinct non-elements of B B. Let be the unique partial order of S whose restriction to B 2 is B, such that b c for all b, c B and conditions (ii), (iv) and (v) of Definition 5.1 hold. As (i) and (iii) obviously hold too, we may define the reflection R(B) of B to be the resulting skew reflection S (B). This definition is essentially due to Meyer; see [20] or [1, pp. 371 373].

VARIETIES OF DE MORGAN MONOIDS: COVERS OF ATOMS 19 By Theorem 5.2, every Dunn monoid B is an RL subreduct of its reflection R(B), and R(B) satisfies e f (by definition) and is distributive (as B is), so R(B) M, by Corollary 5.8. Conversely, the RL reduct of an algebra from M is of course a Dunn monoid, whence so are its subalgebras. This justifies a variant of the Crystallization Fact of [26, p. 124]: Theorem 6.2. The variety of Dunn monoids coincides with the class of all RL subreducts of members of M. Corollary 6.3. The equational theory of M is undecidable. Proof. This follows from Theorem 6.2, because Urquhart [28, p. 1070] proved that the equational theory of Dunn monoids is undecidable. Corollary 6.4. M is not generated (as a variety) by its finite members. Proof. This follows from Corollary 6.3, as M is finitely axiomatized. Clearly, in the statements of Theorem 6.2 and Corollary 6.3, we may replace M by any variety K such that M K DMM. The same applies to Corollary 6.4 if K is also finitely axiomatized. In particular, the variety U is not generated by its finite members. The notational conventions of Definition 5.1 are assumed in the next lemma. Lemma 6.5. Let B be a Dunn monoid. (i) If C is a subalgebra of B, then C {c : c C} {0, 1 } is the universe of a subalgebra of R(B) that is isomorphic to R(C), and every subalgebra of R(B) arises in this way from a subalgebra of B. (ii) If θ is a congruence of B, then R(θ) := θ { a, b : a, b θ} { 0, 0, 1, 1 } is a congruence of R(B), and R(B)/R(θ) = R(B/θ). Also, every proper congruence of R(B) has the form R(θ) for some θ Con B. (iii) If {B i : i I} is a family of Dunn monoids and U is an ultrafilter over I, then i I R(B i)/u = R ( i I B i/u ). Proof. The first assertions in (i) and (ii) are straightforward. For the final assertions, one shows that if D is a subalgebra and ϕ a proper congruence of R(B), then D is the reflection of the subalgebra of B on D B, while ϕ = R(B 2 ϕ). To see that ϕ R(B 2 ϕ), observe that if ϕ identifies a with b (a, b B), and therefore a with b, it must identify 1 = a b with b a B. But this contradicts Lemma 2.1(i), because R(B) is rigorously compact. (iii) For each i I, let 0 i and 1 i denote the extrema of R(B i ) and, for convenience, define 0 i = {0 i } and 1 i = {1 i } and (B ) i = B i. By 0, 1, we mean (for the moment) the extrema of R ( i I B i/u ). Consider x i I R(B i). As U is an ultrafilter, there is a unique F (x) {B, B, 0, 1 } such that {i I : x(i) F (x) i } U

20 T. MORASCHINI, J.G. RAFTERY, AND J.J. WANNENBURG (see [6, Cor. IV.3.13(a)]). If F (x) is 0 [resp. 1 ], define h(x) to be 0 [resp. 1 ]. If F (x) = B, define h(x) = z/u, where z i I B i and, for each i I, { x(i) if x(i) Bi ; z(i) = e B i otherwise. If F (x) = B, define h(x) = (z/u), where z i I B i and, for each i I, { the unique b Bi such that x(i) = b, if this exists; z(i) = e B i, otherwise. Then h is a homomorphism from i I R(B i) onto R ( i I B i/u ), whose kernel is { x, y ( i I R(B i) ) 2 : {i I : x(i) = y(i)} U}, so the result follows from the Homomorphism Theorem. Definition 6.6. Given a variety K of Dunn monoids, the reflection R(K) of K is the subvariety V{R(B) : B K} of M. As a function from the lattice of varieties of Dunn monoids to the subvariety lattice of M, the operator R is obviously isotone. Lemma 6.7. R is order-reflecting and therefore injective. Proof. Let R(K) R(L), where K and L are varieties of Dunn monoids. We must show that K L. Let A K be SI. It suffices to show that A L. By assumption, R(A) R(L). Also, R(A) is SI (because A is), so by Jónsson s Theorem, R(A) HSP U {R(B) : B L}. Because L is closed under H, S and P U, it follows from Lemma 6.5 that R(A) = R(B) for some B L, whence A = B, and so A L. A Brouwerian algebra is an RL satisfying x y = x y, or equivalently, a Dunn monoid satisfying x e. Every variety of countable type has at most 2 ℵ 0 subvarieties, and it is known that there are 2 ℵ 0 distinct varieties of Brouwerian algebras [29]. So, the injectivity of R in Lemma 6.7 yields the following conclusion. Theorem 6.8. The variety M has 2 ℵ 0 7. Covers of Atoms distinct subvarieties. When a lattice L has a least element, its atoms are the covers of. Provided that L is modular, the join of any two distinct atoms covers each join-and, so a cover c of an atom is interesting when it is not the join of two atoms. If L is distributive, that is equivalent to the ostensibly stronger demand that c be join-irreducible. The lattice of subvarieties of a congruence distributive variety E is itself distributive [16, Cor. 4.2]. Therefore, once the atoms of this lattice have been determined, the immediate concern is to identify the join-irreducible covers of each atom E ; we refer to these as covers of E within E. In particular, it behoves us to investigate the join-irreducible covers, within DMM, of the four varieties in Theorem 3.2.